# Deriving the Formula of the Vertex of Quadratic Functions

In getting the vertex of the quadratic function in general form , we usually need to convert it to the vertex form . In the latter form, the vertex of the parabola is at . For example, the function in the general form

can be rewritten in the vertex form as

.

In the vertex form, it is easy to see that the vertex is at .

Aside from this method, we can also use the ordered pair

in place of .

This means that we can get the vertex of without changing the general form to vertex from. In this post, we are going to derive the ordered pair above. That is, we are going to show that

and

.

Expanding gives us

.

Now, the terms with both x’s in the general and vertex forms are and . So, we can equate the two expressions and solve for .

.

Now, we can use as the value of in the general from. That is

.

This means that we can get the vertex of the function

without converting it to the vertex form as follows.

So, the vertex is at which is consistent with our calculation above.